arXiv subjects

A. R. Humphries

Publications and source records attributed to A. R. Humphries.

5 recordsLinked to original sources

arXiv2025

Practicalities of State-Dependent and Threshold Delay Differential Equations

Delays are ubiquitous in applied problems, but often do not arise as the simple constant discrete delays that analysts and numerical analysts like to treat. In this chapter we show how state-dependent delays arise naturally when modeling and the consequences that follow. We treat discrete state-dependent delays, and delays implicitly defined by threshold conditions. We will consider modeling, formulation as dynamical systems, linearization, and numerical techniques. For discrete state-dependent delays we show how breaking points can be tracked efficiently to preserve the order of numerical methods for simulating solutions. For threshold conditions we will discuss how a velocity ratio term arises in models, and present a heuristic linearization method that avoids Banach spaces and sun-star calculus, making the method accessible to a wider audience. We will also discuss numerical implementations of threshold and distributed delay problems which allows them to be treated numerically with standard software.

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arXiv2020

Lyapunov-Razumikhin techniques for state-dependent delay differential equations

We present Lyapunov stability and asymptotic stability theorems for steady state solutions of general state-dependent delay differential equations (DDEs) using Lyapunov-Razumikhin methods. Our results apply to DDEs with multiple discrete state-dependent delays, which may be nonautonomous for the Lyapunov stability result, but autonomous (or periodically forced) for the asymptotic stability result. Our main technique is to replace the DDE by a nonautonomous ordinary differential equation (ODE) where the delayed terms become source terms in the ODE. The asymptotic stability result and its proof are entirely new, and based on a contradiction argument together with the Arzela-Ascoli theorem. This approach alleviates the need to construct auxiliary functions to ensure the asymptotic contraction, which is a feature of all other Lyapunov-Razumikhin asymptotic stability results of which we are aware. We apply our results to a state-dependent model equation which includes Hayes equation as a special case, to directly establish asymptotic stability in parts of the stability domain along with lower bounds on the size of the basin of attraction.

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arXiv2017

Resonance phenomena in a scalar delay differential equation with two state-dependent delays

We study a scalar DDE with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three only contain constant delays. We implemented this expansion and the computation of the normal form coefficients in Matlab using symbolic differentiation. Numerical continuation of the torus bifurcation curves confirms the correctness of our normal form calculations. Moreover, it enables us to compute the curves of torus bifurcations more globally, and to find associated curves of saddle-node bifurcations of periodic orbits that bound the resonance tongues. The tori themselves are computed and visualized in a three-dimensional projection, as well as the planar trace of a suitable Poincaré section. In particular, we compute periodic orbits on locked tori and their associated unstable manifolds. This allows us to study transitions through resonance tongues and the breakup of a 1:4 locked torus. The work presented here demonstrates that state dependence alone is capable of generating a wealth of dynamical phenomena.

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arXiv2017

Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations

We present generalised Lyapunov-Razumikhin techniques for establishing global asymptotic stability of steady-state solutions of scalar delay differential equations. When global asymptotic stability cannot be established, the technique can be used to derive bounds on the persistent dynamics. The method is applicable to constant and variable delay problems, and we illustrate the method by applying it to the state-dependent delay differential equation known as the sawtooth equation, to find parameter regions for which the steady-state solution is globally asymptotically stable. We also establish bounds on the periodic orbits that arise when the steady-state is unstable. This technique can be readily extended to apply to other scalar delay differential equations with negative feedback.

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arXiv2015

Periodic Solutions of a Singularly Perturbed Delay Differential Equation With Two State-Dependent Delays

Periodic orbits and associated bifurcations of singularly perturbed state-dependent delay differential equations (DDEs) are studied when the profiles of the periodic orbits contain jump discontinuities in the singular limit. A definition of singular solution is introduced which is based on a continuous parametrisation of the possibly discontinuous limiting solution. This reduces the construction of the limiting profiles to an algebraic problem. A model two state-dependent delay differential equation is studied in detail and periodic singular solutions are constructed with one and two local maxima per period. A complete characterisation of the conditions on the parameters for these singular solutions to exist facilitates an investigation of bifurcation structures in the singular case revealing folds and possible cusp bifurcations. Sophisticated boundary value techniques are used to numerically compute the bifurcation diagram of the state-dependent DDE when the perturbation parameter is close to zero. This confirms that the solutions and bifurcations constructed in the singular case persist when the perturbation parameter is nonzero, and hence demonstrates that the solutions constructed using our singular solution definition are useful and relevant to the singularly perturbed problem. Fold and cusp bifurcations are found very close to the parameter values predicted by the singular solution theory, and we also find period-doubling bifurcations as well as periodic orbits with more than two local maxima per period, and explain the alignment between the folds on different bifurcation branches.

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